Arccos Calculator

What arccos returns

Cosine starts with an angle and reports a ratio. Arccos, also called acos or inverse cosine, starts with the ratio and walks back to an angle. The ratio is adjacent side divided by hypotenuse in a right triangle, so arccos(0.5) is 60 degrees: the adjacent side is half the hypotenuse at that angle.

Use the NIST Digital Library of Mathematical Functions for the formal definition and principal-value range. The Wolfram MathWorld inverse cosine reference gives the same range and useful identities. On a calculator, the key may say acos, cos^-1, or shift-cos. The superscript minus one means inverse function, not one divided by cosine.

The input is limited because two sides of a right triangle cannot make a ratio outside -1 to 1. Values 1.2 and -1.4 are not large angles. They signal a wrong side pairing, a unit mistake, or rounded measurements that no longer form a possible ratio.

The principal angle is 0 to 180 degrees

Cosine repeats, so one ratio belongs to many angles. A cosine of 0.5 occurs at 60 degrees, 300 degrees, -60 degrees and every full turn added to those positions. Arccos chooses the one answer in the closed interval from 0 through 180 degrees. That is called the principal value, and it makes a function with one dependable output.

The sign tells you which half of the circle contains the principal result. A positive cosine gives an angle from 0 to 90 degrees. A negative cosine gives an angle from 90 to 180 degrees. At zero, the answer is exactly 90 degrees. If a drawing needs the angle in the lower half of the circle, use the principal result as a reference and choose the branch from the geometry.

For example, arccos(-0.5) is 120 degrees, while 240 degrees has the same cosine. Neither is more correct without a drawing; arccos selects 120 degrees by definition. Navigation and robotics usually use atan2 when both coordinate signs matter, because one ratio alone cannot identify a full direction.

Degrees, radians, and calculator mode

The answer can be displayed in degrees or radians. Convert radians to degrees by multiplying by 180 and dividing by pi. Convert degrees to radians by multiplying by pi and dividing by 180. The anchor points are 60 degrees = 1.0472 radians, 90 degrees = 1.5708 radians, and 180 degrees = 3.1416 radians.

JavaScript Math.acos, Python math.acos, and most programming libraries return radians. If a program reports 1.0472 for arccos(0.5), it has not disagreed with the 60 degree answer; it has used the other unit. Handheld calculators also have DEG and RAD modes. A mode mismatch can turn 60 degrees into a displayed value near 1.0472, or make a radian input look like an enormous angle.

Near the endpoints, small ratio errors can have different angle effects. Around cosine 0, the curve changes steadily. Near cosine 1, a small measurement change can move the angle noticeably because the curve is flat at the top. Keep extra digits in the side ratio until the final rounding.

Right triangles: adjacent over hypotenuse

For a right triangle, label the angle first. The adjacent side touches that angle, but it is not the hypotenuse. The hypotenuse is always the side opposite the 90 degree corner and is the longest side. Then use cos(theta) = adjacent / hypotenuse and take arccos of the quotient.

A triangle with an adjacent side of 6 and hypotenuse of 10 has a ratio of 6 / 10 = 0.6. Arccos(0.6) gives 53.1301 degrees, or 0.9273 radians. A 3 foot adjacent side with a 5 foot hypotenuse has the same ratio and the same angle: scale changes the triangle, not its shape.

A negative adjacent side is useful in coordinate geometry, where signs mark direction. In a physical right triangle, side lengths are normally positive, so a negative input usually means you are describing a signed horizontal change rather than measuring a piece of wood.

Choosing arccos, arcsin, or arctan

Choose the inverse function from the two sides you know. Arccos uses adjacent and hypotenuse. Arcsin uses opposite and hypotenuse. Arctan uses opposite and adjacent. The ratio must be formed with respect to the angle you want, not with respect to whichever two labels are easiest to read.

The three examples describe the same triangle, so they agree. If they do not agree in your work, a side was named from the wrong angle or the hypotenuse was mistaken for a leg.

One useful check is arccos(x) + arcsin(x) = 90 degrees for a valid x. For x = 0.6, arccos(0.6) is 53.1301 degrees and arcsin(0.6) is 36.8699 degrees. Their sum is 90 degrees.

Using arccos for slopes and elevation

Arccos can describe a slope when you know the horizontal run and the sloping length. A roof rafter that runs 12 feet and measures 13 feet has cos(theta) = 12 / 13 = 0.9231, so theta = 22.6199 degrees above horizontal. The complementary angle from vertical is 67.3801 degrees.

For a ramp, use the horizontal distance as the adjacent side and the ramp surface as the hypotenuse. A 4 foot horizontal run and a 5 foot surface gives arccos(0.8) = 36.8699 degrees. This is a geometric angle, not a code approval or accessibility determination. For accessible design requirements, consult the United States Access Board ADA standards and the local authority.

In surveying, a horizontal distance and a line-of-sight distance also form this ratio. Keep all distances in the same unit before dividing. Twelve meters and 20 feet cannot be mixed in one ratio, although converting both to meters first leaves the angle unchanged.

Law of cosines: arccos beyond right triangles

Arccos also solves a general triangle when all three sides are known. For sides a, b, and c, the angle opposite c is arccos((a² + b² - c²) / (2ab)). This is the law of cosines, and it reduces to the right triangle rule when c is the hypotenuse and the angle is 90 degrees.

Take sides a = 5, b = 6, and c = 7. The cosine ratio is (25 + 36 - 49) / (2 x 5 x 6) = 12 / 60 = 0.2, so the angle opposite the 7 side is arccos(0.2) = 78.463 degrees. The OpenStax law of cosines explanation shows why the side opposite the target angle gets the minus sign.

Do not round the ratio too early. Side measurements with only a few significant digits cannot support a long decimal angle, but keeping the full calculator value until the final step prevents extra rounding drift.

Common arccos mistakes

Using the wrong side pair. Opposite over hypotenuse belongs to arcsin. Adjacent over opposite belongs to arctan. Swapping the pair changes the angle.

Entering an angle instead of a ratio. The calculator expects a cosine value such as 0.6, not 60. If you have 60 degrees and want its cosine, use the ordinary cosine key.

Ignoring degree and radian mode. A result of 1.0472 is a radian value for 60 degrees. Label the unit before copying it into another calculation.

Rounding a ratio outside the domain. Measurements may produce 1.0002 after rounding even though the true ratio is 1. Use unrounded measurements when possible, and treat a value just beyond 1 as a measurement issue rather than forcing a result.

Assuming the answer identifies a full direction. Arccos returns one principal angle. A map bearing needs signed coordinates and usually atan2, not a lone cosine ratio.